La tour de fibrations exactes des n-catégories. (French) Zbl 0562.18007

Bourn, Dominique

Cah. Topol. Géom. Différ. 25, 327-351 (1984).

The author studies some aspects of n-categories. First he produces a basic construction on a full subcategory V’ of a left exact category V, reflective by a left exact functor K. The construction produces a full subcategory \(Cat_ K(V)\) of the category Cat(V) of internal categories in V; the objects of \(Cat_ K(V)\) are those internal categories trivialized by K. It turns out that \(Cat_ K(V)\) is again a left exact category and V is a full subcategory of \(Cat_ K(V)\), reflective by a left exact functor \(K_ 1\). Thus the construction can be iterated.
The object of the paper is to prove two results. First, the category of n-categories can be obtained by iterating n times the above construction, starting from \(1\to Sets\). Second, the functor K involved in the construction is actually a fibration; when it is an exact fibration, so is \(K_ 1\). As a consequence the author obtains, for every exact category E, a tower of exact fibrations: \[ \infty -Cat(E)...\to n- Cat(E)\to (n-1)-Cat(E)\to...\to Cat(E)\to E\to 1. \] The author promises to deduce from this, in a further paper, some Dold-Kan theorem for n- categorical abelian groups (equivalence between the category of abelian complexes of length n and the category of internal n-categories in Ab).

Reviewer: F.Borceux

MSC:

18G35	Chain complexes (category-theoretic aspects), dg categories
18D30	Fibered categories
18D05	Double categories, \(2\)-categories, bicategories and generalizations (MSC2010)
18E10	Abelian categories, Grothendieck categories

Keywords:

n-categories; internal categories; exact category; fibration; abelian complexes

References:

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